Hallel in the House of Mourning on Rosh Chodesh

Should a mourner recite Hallel on Rosh Chodesh, and what is the ruling for the other worshippers?

Mishna Berura, Siman 131, S"K 20: "Hallel is not recited in the house of mourning. And on Rosh Chodesh, when praying in the house of mourning, even if they later go home, it is not necessary to recite it, but on Chanukah, when they complete the Hallel, it should be recited later at home. But on Shabbat, when Rosh Chodesh falls, or if they buried the deceased on a festival, even if they pray in the house of mourning, they recite Hallel because there is no mourning."
The Rokeach wrote three reasons why Hallel is not recited in the house of mourning. 1. They do not say words of joy. 2. Because of mocking the poor, as the soul is grieving there, and how can they say "the dead do not praise the Lord." 3. Ten who separated are considered as individuals.
In Shu"T Noda B'Yehuda, Mahadura Tinyana - Yoreh De'ah, Siman 215, it is written that it is customary for the mourner to leave to another room, and the rest of the congregation in his house recite Hallel, because the main reason is the first of the Rokeach, as the second reason is not cited by the poskim. The third reason does not apply to us, as we rule that an individual recites Hallel. And when the mourner is not there, even the first reason does not apply, and words of joy can be said.
In Kitzur Shulchan Aruch, Siman 237, Sif 6, although the opinion of mocking the poor is cited, this solution is written for the mourner to leave the room.
Practically, in Gesher HaChaim, chapter 20, it is written that in the place where the deceased died, others should not recite Hallel because of the reason of mocking the poor.